推 Dawsen:第二種有一般通式的構造法嘛 07/26 23:25
→ cmrafsts:二進位吧，沒仔細想，列出8l+r的所有狀況就可能發現規律 07/27 00:50
推 myflame:不知道我的構造法有沒有回答到 07/31 00:05
※ 引述《FAlin (FA（バルシェ應援）)》之銘言：
: 1. Prove that for any two positive integers k , n there exist positive
: integers m_1 , m_2 , ... , m_k such that
: 2^k - 1 1 1 1
: 1 + ------- = ( 1 + --- )( 1 + --- )...( 1 + --- ) .
: n m_1 m_2 m_k
: 2. Giver 2013 red and 2014 blue points in the plane , no three of them on a
: line. We aim to split plane by lines (not passing through these points)
: into regions such that there are no regions containing points of both the
: colors. What is the least number of lines that always suffice?
: 3. Let ABC be a triangle and that A_1 , B_1 , and C_1 be points of cantact of
: the excircles with the sides BC , AC , and AB , respectively. Prove that if
: the circumcenter of △A_1B_1C_1 lies on the circumcircle of △ABC , then
: △ABC is a right triangle.
--------------------------------------防第一題雷---------------------------
k=1取m1=n
k>1取m1=n，如果n是奇數
m1=n+2^k-2，如果n是偶數
除掉後就可以化為k比較小的狀況，by induction and we are done.
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這題另一個想法是希望那k個分數可以以某種方法通分，使恰有一個分子比分母多2^i
i=1,2,...,k-1
而且可以對消。像k=3 n=4l+1時可以用
4l+2 4l+4 4l+8
------ ------ ------
4l+1 4l+2 4l+4
所以大概可以直接構造m1~mk。
台灣隊兩種方法各有一半人用。
--------------------------------底部防雷----------------------------------
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※ 編輯: cmrafsts 來自: 200.69.102.144 (07/26 21:01)